For example, what is the first Chern class of $X:=\mathbb{P}(T\mathbb{P}^3)$ and $Y:=\mathbb{P}(T^*\mathbb{P}^3)$?

I am asking this question because I saw an essay today by F.Hirzebruch, saying that the number $c_1^5$ of $X$ is 4500, while that of $Y$ is 4860, thus providing an example of two diffeomorphic manifolds with different Chern numbers. He did not give the details, and that's why I am here for help.

Hopefully this is not a too stupid question, and thank you all in advance.